# Radius of curvature of lens: Newtons Ring method vs spherometer

I have to do an experiment to find out the radius of curvature of a lens using the Newtons ring method given that you know the wavelength of the monochromatic light used in the experiment.

What the question is that why we are using the Newtons ring when we have a spherometer available which expects the answer that the measurement by one of them is more accurate than the other( Internet tells me that the newtons ring is more accurate than the spherometer without citing a reason)

So in short what I want to know about is Why is using Newton's Ring method to calculate the radius of curvature of the lens more accurate than the spherometer measurement.

Some of my observation so far

1. I have compared the measurements of the radius of curvature of my friends measurement, and found that the newtons ring and spherometer measurement deviates upto 3cm.
2. When I have looked at the principle behind both of the experiments I have found no approximation in the formulas making the question puzzling

Newton's rings, or any other interferometric surface profiling method, are generally better than spherometer measurements for three reasons:

1. The fringe pattern shows the shape of the whole surface: you can see surface deviations from sphericity and indeed with sophisticated enough data processing, not only can you infer the surface radius, but you can also infer all the co-efficients defining more general, nonspherical surfaces (e.g. a general quadric surface model with two sectional curvatures as well as tilts). The spherometer infers the surface radius from the relationship between only three points: so it can't tell the difference between spherical and ellipsoidal surfaces, for example.
2. Fringes are continuous lines: therefore when you fit smooth curves to them and infer the quadric surface that they must imply, you are combining every point in each fringe to get your surface co-efficient estimates. There is therefore considerable averaging and very high tolerance to a few "dud" pixels: if you imagine drawing "error bars" or "error regions" around the fringe and fitting different fringes consistent with the error regions, you find that quite high fringe position uncertainties imply quite small uncertainty in the inferred surface parameters. Again, the spherometer samples only three points, so there is no such averaging.
3. The spherometer depends on the correct working of all the pressure sensors - there is more to go wrong. Interferometric methods are self correcting and self calibrating.

Lastly, the 3cm error by itself sounds nasty, but needs to be thought of in the context of other lens parameters: it may not be that bad. For example, consider a lens with a curvature radius 200cm and lens diameter of 6cm - consistent with an element in a camera lens. What would an uncertainty of 3cm in curvature radius mean in this case? For a 200cm radius, the surface sag over the 6cm diameter is:

$R - \sqrt{R^2 - \frac{d^2}{4}} = 200 - \sqrt{200^2 - \frac{6^2}{4}} = 0.0225013\mathrm{cm}$

$R - \sqrt{R^2 - \frac{d^2}{4}} = 203 - \sqrt{203^2 - \frac{6^2}{4}} = 0.0221687\mathrm{cm}$
so that the difference is $3.3\times10^{-4}\mathrm{cm}$ or about 3.3 microns: only a few fringes in the interferometric method. It is often more representative, therefore, to speak of deviations between curvature measurements in terms of interferometric fringes rather than differences between radiusses.